Uniform approximation

Remarks

Example: on $S = [0, 1]$, the function $e^x$ can be uniformly approximated by polynomials. By Taylor's theorem, $\abs{e^x - \sum_{k=0}^{n} x^k / k!} \le e / (n + 1)!$ for $x \in [0, 1]$, and the bound tends to $0$.

Non-example: on $S = \R$, the function $e^x$ cannot be uniformly approximated by polynomials. For each polynomial $p$, the difference $e^x - p(x)$ tends to $\infty$ as $x \to \infty$, so the supremum is infinite.

For bounded functions, the quantity $\sup_S \abs{f - g}$ is the distance in the supremum norm, and uniform approximation is function approximation in that norm. Complex-valued functions are treated in the same way.